In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. 6 is a factor of n(n + 1)(n + 2).

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements, typically involving positive integers. It consists of two main steps: the base case, where the statement is verified for the initial value (usually n=1), and the inductive step, where one assumes the statement holds for n=k and then proves it for n=k+1. This method is essential for proving properties that are true for all integers in a specified range.

Factors and Multiples

A factor of a number is an integer that divides that number without leaving a remainder. In this context, the statement asserts that 6 is a factor of the product n(n + 1)(n + 2), which means that this product must be divisible by 6 for all positive integers n. Understanding factors and multiples is crucial for verifying divisibility in algebraic expressions.

Properties of Consecutive Integers

The expression n(n + 1)(n + 2) involves three consecutive integers: n, n + 1, and n + 2. One important property of any three consecutive integers is that at least one of them is divisible by 2, and at least one is divisible by 3. This property is key to proving that their product is divisible by 6, as 6 is the product of these two prime factors.